Permutable complements: Difference between revisions

Revision as of 08:58, 30 May 2007

This article defines a symmetric relation on the collection of subgroups inside the same group.

Definition

Symbol-free definition

Two subgroup of a group are said to be permutable complements if:

Their intersection is trivial
Their product is the whole group

Definition with symbols

Two subgroups $H$ and $K$ of a group $G$ are termed permutable complements if the following two conditions hold:

$H \cap K$ is the trivial group
$H K = G$

Facts

Permutable complements need not be unique

Given a subgroup $H$ of $G$ , there may or may not exist permutable complements of $H$ . Moreover, there may exist multiple possibilities for a complement to $H$ , and the multiple possibilities may not even be pairwise isomorphic.

Further information: Every group of given order is a permutable complement for symmetric groups

For a normal subgroup, they are fixed upto isomorphism

Interestingly, when a subgroup is normal, then any two permutable complements to it must be isomorphic. In fact, any permutable complement to it must be isomorphic to the quotient group.

@@ Line 23: / Line 23: @@
 Given a subgroup <math>H</math> of <math>G</math>, there may or may not exist permutable complements of <math>H</math>. Moreover, there may exist multiple possibilities for a complement to <math>H</math>, and the multiple possibilities may not even be pairwise isomorphic.
-{{further|[[Every subgroup of given order is a permutable complement for symmetric groups]]}}
+{{further|[[Every group of given order is a permutable complement for symmetric groups]]}}
 ===For a normal subgroup, they are fixed upto isomorphism===
 Interestingly, when a subgroup is [[normal subgroup|normal]], then any two permutable complements to it must be isomorphic. In fact, any permutable complement to it must be isomorphic to the [[quotient group]].